lyOGARITHMS. 213 



32. A Logarithmic Series Convergent for Integrai. 



\^Ai.uES OF X. In the logarithmic series 



234 



Substitute — .v for x and we shall have 



Jf^ x^ X* 

 \oge(i—x)=—X _--_ ... (2) 



Subtracting (2) from (j), observing that logeTi -{-x)—\oge(i—x) 



loge ,, we obtain 



1 i+x 

 log,--__^:=2 



•^■+|-^'+j-^-=+5--^"+ • • • I (3) 



I 22'-f-2 22" 



Now put A== -—, whence i-\-x= — ; — , i—x= , and 



25 -f I 22'-}- I 2.2' -f I 



""= - . Therefore we obtain 

 I— .V 2. 



Whence, since loge /^=logtYi i-rj— loge^, by substituting and 

 transposing log (.2 we have 



This series converges rapidly for integral values of 2. Its use 

 in computing the logarithms of numbers will now be explained. 



33. To Compute THE Naperian Logarithms of Numbers. 

 The logarithm of i is o in all systems. To compute loge2, put 

 ' = 1 in equation (z) above. We then obtain 



Now put 2 = 2 in equation (^). Then we have 



To find log (.4 we know loge4=loge2^=2 logo 2 ; whence 

 loge4 = i. 3862944 



